willalso be denoted byγ∗. Whenγis one to one on [a,b) and continuous on

[a,b]

we callγ∗asimplecurve.

The idea is that it makes sense to talk of the length of the curve γ

([a,b])

, defined as V

(γ,[a,b])

. For this
reason, in the case that γ is continuous, such an image of a bounded variation function is called a
rectifiable curve.

Definition 4.1.2Letγ :

[a,b]

→ ℝpbe of bounded variation and let f : γ∗→ ℝp. Letting P≡

{t0,⋅⋅⋅,tn}

where a = t0< t1<

⋅⋅⋅

< tn = b, define

∥P ∥ ≡ max {|tj − tj−1| : j = 1,⋅⋅⋅,n}

and the Riemann Steiltjes sum by

∑n
S (P ) ≡ f (γ (τj))⋅(γ(tj)− γ(tj− 1))
j=1

where τj∈

[tj−1,tj]

. (Note this notation is a little sloppy because it does not identify the specific point,τjused. It is understood that this point is arbitrary.) Define∫γf ⋅ dγas the unique numberwhich satisfies the following condition. For all ε > 0 there exists a δ > 0 such that if

∥P∥

≤ δ,then

|∫ |
|| f ⋅dγ − S(P)||< ε.
|γ |

Sometimes this is written as

∫
f ⋅dγ ≡ lim S(P ).
γ ∥P∥→0

The set of points in the curve,γ

([a,b])

will be denoted sometimes byγ∗. Also, when convenient, I willwrite∑Pto denote a Riemann sum.

Then γ∗ is a set of points in ℝp and as t moves from a to b,γ

(t)

moves from γ

(a)

to γ

(b)

. Thus γ∗
has a first point and a last point.

Note that from the above definition, it is obvious that the line integral is linear. Simply let Pn refer to a
uniform parition of